Properties

Label 171.4.f.h
Level 171171
Weight 44
Character orbit 171.f
Analytic conductor 10.08910.089
Analytic rank 00
Dimension 1212
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,4,Mod(64,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 171=3219 171 = 3^{2} \cdot 19
Weight: k k == 4 4
Character orbit: [χ][\chi] == 171.f (of order 33, degree 22, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 10.089326611010.0893266110
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x12+50x10+1797x8+29198x6+345409x4+2092128x2+8856576 x^{12} + 50x^{10} + 1797x^{8} + 29198x^{6} + 345409x^{4} + 2092128x^{2} + 8856576 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 223 2^{2}\cdot 3
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β6+β4+9β29)q4+(β10+β1)q5+(β11β45)q7+(β8+β56β3)q8+(5β11+5β9+8)q10++(40β10+10β5238β1)q98+O(q100) q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} + 9 \beta_{2} - 9) q^{4} + (\beta_{10} + \beta_1) q^{5} + (\beta_{11} - \beta_{4} - 5) q^{7} + ( - \beta_{8} + \beta_{5} - 6 \beta_{3}) q^{8} + ( - 5 \beta_{11} + 5 \beta_{9} + \cdots - 8) q^{10}+ \cdots + ( - 40 \beta_{10} + 10 \beta_{5} - 238 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q52q460q756q10+178q13172q16+296q19944q22458q25140q28+196q31400q34+628q37+48q40+642q43960q462296q49+1180q97+O(q100) 12 q - 52 q^{4} - 60 q^{7} - 56 q^{10} + 178 q^{13} - 172 q^{16} + 296 q^{19} - 944 q^{22} - 458 q^{25} - 140 q^{28} + 196 q^{31} - 400 q^{34} + 628 q^{37} + 48 q^{40} + 642 q^{43} - 960 q^{46} - 2296 q^{49}+ \cdots - 1180 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12+50x10+1797x8+29198x6+345409x4+2092128x2+8856576 x^{12} + 50x^{10} + 1797x^{8} + 29198x^{6} + 345409x^{4} + 2092128x^{2} + 8856576 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (421097ν10+18574850ν8+667580109ν6+9091814878ν4++777219275616)/511964824416 ( 421097 \nu^{10} + 18574850 \nu^{8} + 667580109 \nu^{6} + 9091814878 \nu^{4} + \cdots + 777219275616 ) / 511964824416 Copy content Toggle raw display
β3\beta_{3}== (421097ν1118574850ν9667580109ν79091814878ν5+265254451200ν)/511964824416 ( - 421097 \nu^{11} - 18574850 \nu^{9} - 667580109 \nu^{7} - 9091814878 \nu^{5} + \cdots - 265254451200 \nu ) / 511964824416 Copy content Toggle raw display
β4\beta_{4}== (2500ν1089850ν83229209ν617270450ν4104606400ν2+5014036725)/516093573 ( -2500\nu^{10} - 89850\nu^{8} - 3229209\nu^{6} - 17270450\nu^{4} - 104606400\nu^{2} + 5014036725 ) / 516093573 Copy content Toggle raw display
β5\beta_{5}== (2500ν1189850ν93229209ν717270450ν5104606400ν3+7594504590ν)/516093573 ( -2500\nu^{11} - 89850\nu^{9} - 3229209\nu^{7} - 17270450\nu^{5} - 104606400\nu^{3} + 7594504590\nu ) / 516093573 Copy content Toggle raw display
β6\beta_{6}== (4678649ν10226641250ν88145486525ν6137428566526ν4+9483250101600)/511964824416 ( - 4678649 \nu^{10} - 226641250 \nu^{8} - 8145486525 \nu^{6} - 137428566526 \nu^{4} + \cdots - 9483250101600 ) / 511964824416 Copy content Toggle raw display
β7\beta_{7}== (28392391ν11+2062931550ν9+74141759907ν7+1557563627618ν5++86318337156768ν)/3071788946496 ( 28392391 \nu^{11} + 2062931550 \nu^{9} + 74141759907 \nu^{7} + 1557563627618 \nu^{5} + \cdots + 86318337156768 \nu ) / 3071788946496 Copy content Toggle raw display
β8\beta_{8}== (3392067ν11+159757750ν9+5741693535ν7+91443820458ν5++6684673239840ν)/255982412208 ( 3392067 \nu^{11} + 159757750 \nu^{9} + 5741693535 \nu^{7} + 91443820458 \nu^{5} + \cdots + 6684673239840 \nu ) / 255982412208 Copy content Toggle raw display
β9\beta_{9}== (28392391ν10+2062931550ν8+74141759907ν6+1557563627618ν4++86318337156768)/1535894473248 ( 28392391 \nu^{10} + 2062931550 \nu^{8} + 74141759907 \nu^{6} + 1557563627618 \nu^{4} + \cdots + 86318337156768 ) / 1535894473248 Copy content Toggle raw display
β10\beta_{10}== (64850ν112330709ν973443810ν7447995473ν5++28135631979ν)/3096561438 ( - 64850 \nu^{11} - 2330709 \nu^{9} - 73443810 \nu^{7} - 447995473 \nu^{5} + \cdots + 28135631979 \nu ) / 3096561438 Copy content Toggle raw display
β11\beta_{11}== (64850ν102330709ν873443810ν6447995473ν42713490016ν2+28135631979)/1548280719 ( - 64850 \nu^{10} - 2330709 \nu^{8} - 73443810 \nu^{6} - 447995473 \nu^{4} - 2713490016 \nu^{2} + 28135631979 ) / 1548280719 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6+β4+17β217 \beta_{6} + \beta_{4} + 17\beta_{2} - 17 Copy content Toggle raw display
ν3\nu^{3}== β8+β522β3 -\beta_{8} + \beta_{5} - 22\beta_{3} Copy content Toggle raw display
ν4\nu^{4}== 3β940β6377β2 -3\beta_{9} - 40\beta_{6} - 377\beta_{2} Copy content Toggle raw display
ν5\nu^{5}== 40β86β7+577β3577β1 40\beta_{8} - 6\beta_{7} + 577\beta_{3} - 577\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 150β111297β4+9875 150\beta_{11} - 1297\beta_{4} + 9875 Copy content Toggle raw display
ν7\nu^{7}== 300β101297β5+16360β1 300\beta_{10} - 1297\beta_{5} + 16360\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 5391β11+5391β9+39706β6+39706β4+279311β2279311 -5391\beta_{11} + 5391\beta_{9} + 39706\beta_{6} + 39706\beta_{4} + 279311\beta_{2} - 279311 Copy content Toggle raw display
ν9\nu^{9}== 10782β1039706β8+10782β7+39706β5477841β3 -10782\beta_{10} - 39706\beta_{8} + 10782\beta_{7} + 39706\beta_{5} - 477841\beta_{3} Copy content Toggle raw display
ν10\nu^{10}== 173028β91192549β68145377β2 -173028\beta_{9} - 1192549\beta_{6} - 8145377\beta_{2} Copy content Toggle raw display
ν11\nu^{11}== 1192549β8346056β7+14108122β314108122β1 1192549\beta_{8} - 346056\beta_{7} + 14108122\beta_{3} - 14108122\beta_1 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/171Z)×\left(\mathbb{Z}/171\mathbb{Z}\right)^\times.

nn 2020 154154
χ(n)\chi(n) 11 1+β2-1 + \beta_{2}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
64.1
−2.72529 4.72034i
−1.71809 2.97581i
−1.45636 2.52249i
1.45636 + 2.52249i
1.71809 + 2.97581i
2.72529 + 4.72034i
−2.72529 + 4.72034i
−1.71809 + 2.97581i
−1.45636 + 2.52249i
1.45636 2.52249i
1.71809 2.97581i
2.72529 4.72034i
−2.72529 4.72034i 0 −10.8544 + 18.8003i −1.48079 2.56480i 0 6.79546 74.7206 0 −8.07113 + 13.9796i
64.2 −1.71809 2.97581i 0 −1.90364 + 3.29720i 6.83777 + 11.8434i 0 −20.1525 −14.4069 0 23.4957 40.6958i
64.3 −1.45636 2.52249i 0 −0.241981 + 0.419123i −10.1021 17.4973i 0 −1.64299 −21.8921 0 −29.4246 + 50.9649i
64.4 1.45636 + 2.52249i 0 −0.241981 + 0.419123i 10.1021 + 17.4973i 0 −1.64299 21.8921 0 −29.4246 + 50.9649i
64.5 1.71809 + 2.97581i 0 −1.90364 + 3.29720i −6.83777 11.8434i 0 −20.1525 14.4069 0 23.4957 40.6958i
64.6 2.72529 + 4.72034i 0 −10.8544 + 18.8003i 1.48079 + 2.56480i 0 6.79546 −74.7206 0 −8.07113 + 13.9796i
163.1 −2.72529 + 4.72034i 0 −10.8544 18.8003i −1.48079 + 2.56480i 0 6.79546 74.7206 0 −8.07113 13.9796i
163.2 −1.71809 + 2.97581i 0 −1.90364 3.29720i 6.83777 11.8434i 0 −20.1525 −14.4069 0 23.4957 + 40.6958i
163.3 −1.45636 + 2.52249i 0 −0.241981 0.419123i −10.1021 + 17.4973i 0 −1.64299 −21.8921 0 −29.4246 50.9649i
163.4 1.45636 2.52249i 0 −0.241981 0.419123i 10.1021 17.4973i 0 −1.64299 21.8921 0 −29.4246 50.9649i
163.5 1.71809 2.97581i 0 −1.90364 3.29720i −6.83777 + 11.8434i 0 −20.1525 14.4069 0 23.4957 + 40.6958i
163.6 2.72529 4.72034i 0 −10.8544 18.8003i 1.48079 2.56480i 0 6.79546 −74.7206 0 −8.07113 13.9796i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.4.f.h 12
3.b odd 2 1 inner 171.4.f.h 12
19.c even 3 1 inner 171.4.f.h 12
57.h odd 6 1 inner 171.4.f.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.4.f.h 12 1.a even 1 1 trivial
171.4.f.h 12 3.b odd 2 1 inner
171.4.f.h 12 19.c even 3 1 inner
171.4.f.h 12 57.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(171,[χ])S_{4}^{\mathrm{new}}(171, [\chi]):

T212+50T210+1797T28+29198T26+345409T24+2092128T22+8856576 T_{2}^{12} + 50T_{2}^{10} + 1797T_{2}^{8} + 29198T_{2}^{6} + 345409T_{2}^{4} + 2092128T_{2}^{2} + 8856576 Copy content Toggle raw display
T73+15T72115T7225 T_{7}^{3} + 15T_{7}^{2} - 115T_{7} - 225 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12+50T10++8856576 T^{12} + 50 T^{10} + \cdots + 8856576 Copy content Toggle raw display
33 T12 T^{12} Copy content Toggle raw display
55 T12++448364160000 T^{12} + \cdots + 448364160000 Copy content Toggle raw display
77 (T3+15T2+225)4 (T^{3} + 15 T^{2} + \cdots - 225)^{4} Copy content Toggle raw display
1111 (T65044T4+488138400)2 (T^{6} - 5044 T^{4} + \cdots - 488138400)^{2} Copy content Toggle raw display
1313 (T689T5++3005780625)2 (T^{6} - 89 T^{5} + \cdots + 3005780625)^{2} Copy content Toggle raw display
1717 T12++98 ⁣ ⁣76 T^{12} + \cdots + 98\!\cdots\!76 Copy content Toggle raw display
1919 (T6148T5++322687697779)2 (T^{6} - 148 T^{5} + \cdots + 322687697779)^{2} Copy content Toggle raw display
2323 T12++44 ⁣ ⁣00 T^{12} + \cdots + 44\!\cdots\!00 Copy content Toggle raw display
2929 T12++30 ⁣ ⁣00 T^{12} + \cdots + 30\!\cdots\!00 Copy content Toggle raw display
3131 (T349T2+2385261)4 (T^{3} - 49 T^{2} + \cdots - 2385261)^{4} Copy content Toggle raw display
3737 (T3157T2++28025385)4 (T^{3} - 157 T^{2} + \cdots + 28025385)^{4} Copy content Toggle raw display
4141 T12++75 ⁣ ⁣00 T^{12} + \cdots + 75\!\cdots\!00 Copy content Toggle raw display
4343 (T6++11 ⁣ ⁣25)2 (T^{6} + \cdots + 11\!\cdots\!25)^{2} Copy content Toggle raw display
4747 T12++78 ⁣ ⁣76 T^{12} + \cdots + 78\!\cdots\!76 Copy content Toggle raw display
5353 T12++48 ⁣ ⁣96 T^{12} + \cdots + 48\!\cdots\!96 Copy content Toggle raw display
5959 T12++10 ⁣ ⁣00 T^{12} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
6161 (T6++20 ⁣ ⁣25)2 (T^{6} + \cdots + 20\!\cdots\!25)^{2} Copy content Toggle raw display
6767 (T6++25 ⁣ ⁣25)2 (T^{6} + \cdots + 25\!\cdots\!25)^{2} Copy content Toggle raw display
7171 T12++26 ⁣ ⁣00 T^{12} + \cdots + 26\!\cdots\!00 Copy content Toggle raw display
7373 (T6++20 ⁣ ⁣25)2 (T^{6} + \cdots + 20\!\cdots\!25)^{2} Copy content Toggle raw display
7979 (T6++10 ⁣ ⁣25)2 (T^{6} + \cdots + 10\!\cdots\!25)^{2} Copy content Toggle raw display
8383 (T6+65 ⁣ ⁣24)2 (T^{6} + \cdots - 65\!\cdots\!24)^{2} Copy content Toggle raw display
8989 T12++62 ⁣ ⁣00 T^{12} + \cdots + 62\!\cdots\!00 Copy content Toggle raw display
9797 (T6++1811716000000)2 (T^{6} + \cdots + 1811716000000)^{2} Copy content Toggle raw display
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